It’s been quite a while since my last chaos theory post. I’ve
been caught up in other things, and I’ve needed to do some studying. Based
on a recommendation from a commenter, I’ve gotten another book on Chaos
theory, and it’s frankly vastly better than the two I was using before.

Anyway, I want to first return to dense periodic orbits in chaotic
systems, which is what I discussed in the previous chaos theory
post. There’s a glaring hole in that post. I didn’t so much get itwrong as I did miss the fundamental point.

If you recall, the basic definition of a chaotic system is
a dynamic system with a specific set of properties:

Sensitivity to initial conditions,

Dense periodic orbits, and

topological mixing

The property that we want to focus on right now is the
dense periodic orbits.

In a dynamical system, an orbit isn’t what we typically think of
as orbits. If you look at all of the paths through the phase space of a
system, you can divide it into partitions. If the system enters a state in any
partition, then every state that it ever goes through will be part of the same
partition. Each of those partitions is called an orbit. What
makes this so different from our intuitive notion of orbits is that
the intuitive orbit repeats. In a dynamical system, an
orbit is just a set of points, paths through the phase space of
the system. It may never do anything remotely close to repeating – but it’s
an orbit. For example, if I describe a system which is the state
of an object floating down a river, the path that it takes is
an orbit. But it obviously can’t repeat – the object isn’t going to
go back up to the beginining of the river.

An orbit that repeats is called a periodic orbit. So
our intuitive notion of orbits is really about periodic
orbits.

Periodic orbits are tightly connected to chaotic systems.
In a chaotic system, one of the basic properties is a particular
kind of unpredictability. Sensitivity to initial conditions
is what most people think of – but the orbital property is
actually more interesting.

A chaotic system has dense periodic orbits. Now, what
does that mean? I explained it once before, but I managed to
miss one of the most interesting bits of it.

The points of a chaotic system are dense around
the periodic orbits. In mathematical terms, that means that
every point in the attractor for the chaotic system isarbitrarily close to some point on a periodic orbit. Pick
a point in the chaotic attractor, and pick a distance greater than zero.
No matter how small that distance is, there’s a periodic orbit
within that distance of the point in the attractor.

The last property of the chaotic system – the one which makes
the dense periodic orbits so interesting – is topological mixing. I’m
not going to go into detail about it here – that’s for the next post. But
what happens when you combine topological mixing with the density
around the periodic orbits is that you get an amazing kind of
unpredictability.

You can find stable states of the system, where everything
just cycles through an orbit. And you can find an instance of
the system that appears to be in that stable state. But
in fact, virtually all of the time, you’ll be wrong. The
most miniscule deviation, any unmeasurably small difference between
the theoretical stable state and the actual state of the system – and
at some point, you’re behavior will diverge. You could stay close to the
stable state for a very long time – and then, whammo! the system will
do something that appears to be completely insane.

What the density around periodic orbits means is that
even though most of the points in the phase space aren’t
part of periodic orbits, you can’t possibly distinguish them
from the ones that are. A point that appears to be stableprobably isn’t. And the difference between real stability
and apparent stability is unmeasurably, indistinguishably small.
It’s not just the initial conditions of the system
that are sensitive. The entire system is sensitive. Even if you
managed to get it into a stable state, the slightest pertubation,
the tiniest change, could cause a drastic change at some unpredictable
time in the future.

This is the real butterfly effect. A butterfly flaps its wings –
and the tiny movement of air caused by that pushes the weather system
that tiny bit off of a stable orbit, and winds up causing the
diversion that leads to a hurricane. The tiniest change at any
time can completely blow up.

It also gives us a handle on another property of chaotic systems
as models of real phenomena: we can’t reverse them. Knowing the
measured state of a chaotic system, we cannot tell how it
got there. Even if it appears to be in a stable state, if it’s part
of a chaotic system, it could have just “swung in” the chaotic
state from something very different. Or it could have been in what
appeared to be a stable state for a long time, and then suddenly
diverge. Density effectively means that we can’t distinguish
the stable case from either of the two chaotic cases.

Comments

I think you are using the word “stable” incorrectly. All of the periodic orbits are unstable. Their instability is precisely your point, small perturbations cause trajectories to move away from the orbit. The property of orbits that you are calling “stable” is that they are invariant. If you start exactly on a periodic orbit, you remain on that orbit forever.

Sorry to nitpick, but I think it’s important to clarify that this is a colloquial usage of the word dense rather than a technical mathematical one. (I.e., “X is dense around Y” does not have an accepted mathematical meaning that I know of, although you seem to be using it as a synonymous phrase for the accepted wording “Y is dense in X”. It sounds like you’re saying that X is dense when you really mean that Y is dense.)

However, your description which follows this statement is a perfectly good description of the fact that the periodic orbits are dense.

The textbook I used the first time I was learning about Chaos theory was “A First Course in Chaotic Dynamical Systems” by Robert Devaney (ISBN: 978-0201-554-069). It’s undergraduate level, but well written as I recall.

It also keeps its focus on one dimensional discrete-time systems (i.e. iterating something that maps real numbers to real numbers), although it does cover a few other topics. The advantage of that is the systems can be analyzed explicitly without too much difficulty.

Does that mean that for any periodic orbit, there is a point in that orbit where an arbitrarily small perturbation will derail the system? Can there be regions of the orbit where it is completely stable, or does the denseness also mean that every interval in the orbit has arbitrarily unstable points?

Another nitpick: You seem to be using “partition” in a non-standard way. A partition of a set X is a collection of subsets of X such that each element of X is in exactly one of the subsets. So here the partition would be the collection of orbits, not an orbit.

Well, you don’t seem interested in editing your post to correct any of the issues that have been pointed out in this thread, or the thing about two different notions of tree in the previous Cantor thread. I think these things are pretty important if you want mathematical newbies not to get confused unnecessarily– and I suppose I might be willing to call them glaring errors since, after all, a talented crank can crank out an entire system of crankery on the basis of smaller misunderstandings of technical terms.

Regarding responding to things: see #1 and #8. Not exactly urgent, of course.

I am surprised you haven’t commented about your error in calling the orbits stable. You are a mathematician writing about mathematics and surely you know the importance of using the correct terms for well defined concepts.

I really enjoy this blog. When I read about math that I am not familiar with I always assumed that you got the mathematics right, and when you made the inevitable mistakes you would correct them. If I read these comments without knowing this field, I wouldn’t know if the comments were nonsense or not.

Since you asked, I think you should correct errors in the mathematics, so your readers can be assured they are not learning bad math from this blog.

Actually, jbw and others, the author is using a correct version of the word stable (so are you).

A stable orbit, in one sense, as you pointed out, refers to an orbit when perturbed slightly that trajectories move far away from that orbit.

A slightly different sense would mean something like ‘an orbit around a stable equilibrium point’, i.e. an orbit that is itself a small enough deviation from a constant orbit. By small enough, we mean that we want the points of the orbit itself to remain in a bounded area. This is the only meaning you would meet in a simple classical mechanics class.

In the first sense, we would instead want the points of the nearby orbits to remain close to points of the original orbit; however, neither the original or perturbed orbits would themselves have to be bounded. See ‘floating in a river’ example.

With some mild conditions, the second sense of the word will require a periodic orbit. I’m not sure what the conditions are exactly. A counterexample would be the harmonic oscillator with friction, which is stable in both senses but not periodic (the motion slowly decays.)